Velocity-Time Graphs (v-t graphs)

Theory Exercises

Velocity-Time Graphs (v-t Graphs)

Introduction to v-t Graphs

A velocity-time graph (v-t graph) shows how velocity changes over time.

Key information from v-t graphs:

Vertical axis (y): Velocity (m/s)
Horizontal axis (x): Time (s)
Shape of the line: Shows the type of motion
Slope of the line: Represents acceleration
Area under the line: Represents displacement

Reading and Interpreting v-t Graphs

1. Constant Velocity (Horizontal Line)

Characteristics:

Horizontal line on v-t graph
Slope = 0 (no acceleration)
Uniform rectilinear motion
Velocity remains constant

2. Constant Acceleration (Straight Line with Slope)

Characteristics:

Straight line with non-zero slope
Positive slope = increasing velocity (accelerating)
Negative slope = decreasing velocity (decelerating)
Acceleration is constant

Example (Accelerating):

Example (Decelerating):

3. Variable Acceleration (Curved Line)

Characteristics:

Curved line on v-t graph
Slope changes along the curve
Acceleration is not constant
Acceleration changes over time

Example:

Calculating Properties from v-t Graphs

Finding Acceleration

Acceleration is the slope of the v-t graph.

\[a = \text{slope} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}\]

> [Ejemplo: From a v-t graph, velocity increases from 0 m/s to 20 m/s in 5 seconds. > Acceleration = (20 - 0)/(5 - 0) = 4 m/s²]

Finding Displacement

Displacement is the area under the v-t graph.

\[\text{Displacement} = \text{Area under the curve}\]

For constant velocity (rectangle):

\[\Delta x = v \times t\]

> [Ejemplo: A v-t graph shows constant velocity of 10 m/s for 8 seconds. > Displacement = 10 m/s × 8 s = 80 m]

For constant acceleration (triangle or trapezoid): Triangle (starting from rest):

\[\Delta x = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times t \times v_f\]

> [Ejemplo: Starting from rest, velocity reaches 20 m/s in 10 seconds. > Displacement = ½ × 10 × 20 = 100 m]

Trapezoid (initial velocity ≠ 0):

\[\Delta x = \frac{1}{2}(v_i + v_f) \times t\]

> [Ejemplo: Velocity increases from 5 m/s to 15 m/s over 4 seconds. > Displacement = ½(5 + 15) × 4 = ½ × 20 × 4 = 40 m]

Common Motion Patterns in v-t Graphs

Pattern 1: Starting from Rest and Accelerating

Begins at origin (0, 0)
Straight line with positive slope
Velocity increases uniformly

Pattern 2: Constant Velocity

Horizontal line at some velocity value
No change in velocity
Zero acceleration

Pattern 3: Uniform Deceleration to Rest

Straight line with negative slope
Ends at v = 0
Velocity decreases uniformly

Pattern 4: Changing Direction

Line crosses the time axis (v = 0)
Above axis = motion in positive direction
Below axis = motion in negative direction
Crossing point = momentary rest

> [Ejemplo: An object thrown upward: > - Starts with positive velocity > - Decelerates (negative acceleration) > - Reaches zero velocity at maximum height > - Continues with increasing negative velocity (falling)]

Comparing Different Motions with v-t Graphs

Same Acceleration, Different Initial Velocities

Different Accelerations, Same Time

Solving Problems Using v-t Graphs

Example 1: Finding Displacement from Constant Velocity

Problem: A v-t graph shows velocity of 15 m/s for 8 seconds. Find the displacement. Solution:

Shape: Rectangle
Displacement = area = 15 m/s × 8 s = 120 m

Example 2: Finding Acceleration and Displacement

Problem: A v-t graph shows velocity increasing linearly from 5 m/s to 25 m/s in 10 seconds. Find acceleration and displacement. Solution:

Acceleration: a = (25 - 5)/10 = 20/10 = 2 m/s²
Displacement: Area of trapezoid = ½(5 + 25) × 10 = ½ × 30 × 10 = 150 m

Example 3: Identifying Motion Type from v-t Graph

Problem: Analyze the motion shown by three different v-t graphs. Graph A: Horizontal line at 10 m/s → Uniform motion (a = 0) Graph B: Straight line from (0,0) to (5,20) → Uniform acceleration (a = 4 m/s²) Graph C: Curved line increasing slope → Variable acceleration

Area Under v-t Curve

The area under a v-t curve represents displacement - this is a crucial concept!

Different shapes: Rectangle:

\[A = \text{width} × \text{height} = t × v\]

Triangle:

\[A = \frac{1}{2} × \text{base} × \text{height} = \frac{1}{2} × t × v\]

Trapezoid:

\[A = \frac{1}{2}(b_1 + b_2) × h = \frac{1}{2}(v_i + v_f) × t\]

Relationships Between Kinematic Graphs

Property	x-t graph	v-t graph
Slope represents	Velocity	Acceleration
For constant velocity	Straight line	Horizontal line
For constant acceleration	Parabola	Straight line
For rest	Horizontal line	Point on t-axis
Area under curve	—	Displacement

Real-World Applications

Sports: Analyzing sprinter's acceleration in a race
Traffic: Understanding vehicle acceleration and braking
Physics experiments: Tracking motion of falling objects
Engineering: Designing acceleration profiles for machines
Vehicles: Understanding acceleration and deceleration patterns